ALL POSSIBLE MODEL SELECTION IN PROC MIXED A SAS MACRO APPLICATION
|
|
- Briana Poole
- 6 years ago
- Views:
Transcription
1 Libraries Annual Conference on Applied Statistics in Agriculture th Annual Conference Proceedings ALL POSSIBLE MODEL SELECTION IN PROC MIXED A SAS MACRO APPLICATION George C J Fernandez Follow this and additional works at: Part of the Agriculture Commons, and the Applied Statistics Commons This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License. Recommended Citation Fernandez, George C J (2006). "ALL POSSIBLE MODEL SELECTION IN PROC MIXED A SAS MACRO APPLICATION," Annual Conference on Applied Statistics in Agriculture. This is brought to you for free and open access by the Conferences at. It has been accepted for inclusion in Annual Conference on Applied Statistics in Agriculture by an authorized administrator of. For more information, please contact cads@k-state.edu.
2 Conference On Applied Statistics In Agriculture 147 All Possible Model Selection in PROC MIXED - A SAS Macro Application George C J Fernandez College of Agriculture, Biotechnology and Natural Resources University of Nevada - Reno Reno NV ABSTRACT A user-friendly SAS macro application to perform all possible model selection of fixed effects including quadratic and cross products in the presence of random and repeated measures effects using SAS PROC MIXED is available. This macro application will complement the model selection option currently available in the SAS PROC REG for multiple linear regressions and the experimental SAS procedure GLMSELECT that focuses on the standard independently and identically distributed general linear model for univariate responses. Options are also included in this macro to select the best covariance structure associated with the user-specified fully saturated repeated measures model; to graphically explore and to detect statistical significance of user specified linear, quadratic, interaction terms for fixed effects; and to diagnose multicollinearity, via the VIF statistic for each continuous predictors involved in each model selection step. Two model selection criteria, AICC (corrected Akaike Information Criterion) and MDL (minimal description length) are used in all possible model selection and summaries of the best model selection are compared graphically. The differences in the degree of penalty factors associated with the model dimension between AICC and MDL are investigated. Complete mixed model analysis of final model including data exploration, influential diagnostics, and checking for model violations using the experimental ODS GRAPHICS option available in Version 9.13 is also implemented. Instructions for downloading and running this user-friendly macro application are included. KEYWORDS AICC, Delta AICC, MDL, Repeated Measures, GLMSELECT 1. INTRODUCTION Model selection is usually carried out by the automated procedures built into the software including frequently used forward, backward, and stepwise model selection procedures. There is an extensive review and discussion on the theoretical aspects of model selection criteria and procedures (Burnham and Anderson 2002; Hoeting et.al 2006). However, mixed model selection can be tedious, time consuming and complicated due to the presence of additional random terms and optimal variance-covariance structure associated with repeated measures ( Littell et.al 2006;
3 148 Hoeting et.al 2006; Kramer 2004). Ngo and Rand (2002) developed a SAS macro for performing mixed model selection for user-specified models. Keselman et.al (1998) proposed a SAS based method to select the best covariance structure in mixed model repeated measures analysis. However, to apply these SAS macros in model selection, SAS programming experience is a requirement. Kramer(2004) developed an automated model selection application using SAS Mixed and PERL codes. Programming knowledge in PERL is required to use this application. This paper presents a practical solution for automated and efficient mixed model selection and model exploration using a user-friendly SAS macro application named ALLMIXED. 2. MODEL SELECTION CRITERIA USED IN ALLMIXED MACRO All of the information criteria (IC) are based on the value of 2log L where L is the value of the likelihood function from either the maximum likehood (ML) or the residual maximum likelihood (REML). The general form of the information criteria is IC = -2 log L + Penalty factor (pf), where -2 log L is mainly derived from PROC MIXED method ML. The change in two likelihood functions is ) -2 log L = 2 log L I - 2 log L min. The L ref is the value of the likelihood where -2 log L ref = -2 log L derived from PROC MIXED method ML that contain optional random and repeated measure covariance parameter and user specified Must-Have fixed effects. p = Number of fixed effect terms k = number of random effect terms n = total sample size Information criteria are defined as follows: AIC = -2 log L + 2(p+k+1) ( Hoeting et. al 2006) AIC_C= -2 log L + [2(p+k+1) (n/(n-p-k-2))] ( Hoeting et. al 2006) * In large sample AIC and AIC_C are nearly equivalent )AIC_C = AIC_Ci- AIC_C min AICC sas = AICC reported by SAS PROC Mixed using ML AICC REML = AICC reported by SAS PROC Mixed using REML MDL =1/2 {-2 log L + [log(n) (p+k+1)]} ( Hoeting et. al 2006) ) MDL = MDLi- MDL min
4 Conference On Applied Statistics In Agriculture 149 BIC = -2 log L + [log(n) (p+k+1)] (SAS Institute 2006) Penalty factor % = (pf i / -2 log L ref ) * SAMPLE DATA A sample data set is used to demonstrate the use of the ALLMIXED macro which was simulated with 50 subjects and 5 repeated (Time) measures / subject. The covariance structure is AR(1). Four levels of significant treatments are randomly assigned to 50 subjects with significant covariates : X5, X5 2, X15, X5*X15 and non-significant covariates: X1, X2, X3, X4, X6, X7, X8, X9, X10, X11, X12, X13, X14, X16, X17, X18, X19, X ALL POSSIBLE MODEL SELECTION STEPS The recommended selection steps for performing the model selection in MIXED model is illustrated in Figure1. Although the recommended sequence of the steps are identified in the figure, it is not a requirement to follow the same sequence. Users are free to choose to run any model selection steps in any order the desire. However, before running these model selection steps the data must be prepared suitable for running the SAS PROC Mixed (Littell et.al 2006) procedure. The following types of PC data formats can be used with the ALLMIXED macro: SAS temporary and permanent data files, Microsoft excel or ACCESS data tables, COMMA or TAB delimited text file. Refer to the ALLMIXED macro help file available from the author s website for more information regarding inputting data file name and its location in the macrocall window. Step1: Pre-screening If the number of fixed effects exceeds 15, running all possible models will take very long time to complete. Therefore, under these circumstances, pre-screening is recommended. In prescreening, we can discard the least important model terms and only select the user-specified number of effects based on the LASSO (Tibshirani 1996) and LAR (Efron et al. 2004) methods implemented in the GLMSELECT (Cohen 2006) the experimental SAS procedure available in SAS version For more information of the theory and selection features refer to SAS Institute (2006). Figures 2 and 3 show the summary table of 10 selected fixed effects from a total of 20 potential fixed effects contained in the simulated data based on LASS and LAR respectively. Both methods selected the same continuous (x5 x14 x15 x16 x17 ) and categorical contrasts ( TRT TIME) as important fixed effects in the pre-screening. These fixed effects will be used in the subsequent model selection steps. Refer to the ALLMIXED macro help file available from the authors website for more information regarding inputting appropriate parameters in the macro-call window.
5 150 Step2: Repeated measures - Initial covariance type selection. In a repeated measures modeling, the best covariance structure describing the correlation among the repeated measures should be identified first. The best covariance structure can be identified for different user-specified covariance structures by comparing the AICC statistic computed in PROC MIXED using REML method and select the covariance type which gives the smallest AICC value. In this example, four different covariance types, CS, AR(1), TOEP, and UN are compared in the full saturated model containing two categorical effects (TRT and TIME) and five continuous fixed effects selected from the previous pre-screening steps. The results of initial covariance type selection are graphically displayed in Figure4 and based on )AICC j (AICC j - AICC min), AR(1) can be identified as the best covariance type. Therefore, AR(1) covariance type will be used in the subsequent fixed effect selection. Refer to the ALLMIXED macro help file available from the authors website for more information regarding inputting appropriate parameters in the macro-call window. Step3: All possible model selection steps All combinations of models associated with the user-specified fixed effects are generated by the ALLMIXED macro and their information criteria statistics, AIC_C and MDL are compared in this step. Users can optionally differentiate certain fixed effects as MUST HAVE and other fixed effects as SELECTABLE in the all possible model selection. All combinations of mixed models using the fixed effects listed in SELECTABLE category are generated in this step and the following statistics are estimated. Variance inflation statistics (VIF) for each continuous predictor variable in the model. Information criteria estimates based on REML: AICC reml Information criteria estimates based on ML: AIC, AIC_C, AICC sas, MDL, and BIC. The relationships between AIC, AIC_C, AICC sas, AICC reml, MDL, and BIC are investigated by the rank correlations using the SAS PROC CORR and scatter plot matrix (Figure 5). Perfect rank correlations (1) were observed between (AIC_C and AICC sas ) and (MDL and BIC) indicating that these two sets of IC behave identically in the model selection. Furthermore AIC and AIC_C behaved very similarly and the degrees of penalty were similar when the number of fixed effects is relatively small (p=19) compared with total number of observations (n=250) in this simulated data. The rank correlations between AICC reml and AIC statistics computed by ML method (AIC, AIC_C, AICC sas ) were not perfectly correlated and AICC reml behave differently from ML based IC. Big differences were observed in the model selection performance of AIC based ( AIC, AIC_C, ) and BIC based (MDL) information criteria as evident by the rank correlation. All IC statistics reported here are made out of two components: a) Log likelihood estimate (-2 log L ) b) penalty factor (pf). For a given model, log L value is constant and influenced by degree of model fit, variables included and not included in the model, presence of influential outliers, model specification errors, and severe multicollinarity. The penalty factor is made out of number of fixed (p) and random effects (k) and the sample size (n). When the all possible model
6 Conference On Applied Statistics In Agriculture 151 selection process involving only the fixed effects is carried out, the sample size and the number of random effects become constant. Therefore, only the number of fixed effects becomes the determining component of the penalty factor. The relationship between the penalty factor and the number of fixed effects between AIC_C, AICC reml, and MDL are shown in Figure 6. The penalty factor for the AICC reml becomes constant because this penalty factor does not include any fixed effects and only the number of random effects (which is a constant) is included. The penalty factors for AIC_C and MDL shows a positive linear effect associated with the increase in the number of fixed effects. The MDL penalty factor % increased from 6.5% (Reference model) to about 8.5% where as the AIC_C penalty factor % increased from 2.5% (Reference model) to about 3.2% when the number of fixed effects increased by 5 terms. Thus, the degree of penalty is about 3 times stronger for MDL than the AIC_C and this clearly evident in the ratio between MDL penalty % and AIC_C penalty and the relationship is clearly shown in Figure 6. This findings clearly confirm the earlier reports (Hoeting et. al 2006 ) that MDL statistic favors more parsimonious model than AIC_C based model selection. The components of AIC_C and MDL are graphically compared in Figure 7. For a given model, log L value is constant in the estimation of AIC_C and MDL and it decreases linearly with an increase in the number of fixed terms. But, within a subset (two, three, four variable subset), the log L value varies a lot whereas all the models within a subset have the same penalty factor for both AIC_C and MDL (Figure 7). Furthermore, MDL statistic favors parsimonious model (1,2 and 3 subsets) whereas AIC_C statistic favors models with large number of model terms (3,4,5 subsets) especially in a large data set (50 subjects by 5 repeated measures) (Figure 7). Graphical display of best models within each subset based on smallest )AIC_C within each subset and the list of best candidate models ()AIC_C # 2) are shown in Figure 8. The full model with all five fixed effects was identified as the overall best model and 1, 3-variable model, 3, 4- variable models and 1, 5-variable full models were identified as the best candidate models based on )AIC_C # 2. The log L value always decreases with an increase in number of model terms. Graphical display of best models within each subset based on smallest )MDL within each subset and the list of best candidate models ()MDL # log(n)/2) are shown in Figure9. The 3-variable model was identified as the overall best model and 1, 2-variable model, 4 3-variable models and 3, 4-variable full models were identified as the best candidate models based on ()MDL # log (n)/2). The model selection results for this simulated large data (50 subjects, 5 repeated measures) clearly shows that )MDL favors a parsimonious model whereas )AIC_C favors a model with more terms. However, these findings needs to be further verified by testing different types data sets with different sample sizes. Refer to the ALLMIXED macro help file available from the authors website for more information regarding inputting appropriate parameters in the macro-call window. Step4: Graphical exploration for multicollinarity and model specification error Severe multicollinarity (Variance inflation factor > 10) among predictor variables in mixed model analysis can result in unstable parameter estimates with inflated standard errors. When a
7 152 fixed effect predictor involved in a collinear relationship is dropped from the model, the sign and size of the remaining predictor variable estimates can change dramatically. Therefore, presence of high degree of multicollinarity can impact fixed effect selection. Hence, assessing the degree of multicollinarity for each of the continuous fixed effects in all possible model selection can help to select the best model from the set of best candidate models. Variable(s) not contributing multicollinarity could be preferred over the variables significantly contributing to multicolliniarity. Figure 10 shows the box-plot display of VIF distribution for all the continuous predictors included in model selection. Because the data used in the study are simulated from known properties multicollinarity should not exists and it is clearly shown in Figure10 where VIF values were less than 2 for all the predictor variables. Also, to diagnose multicollinearity in each model selection step (when VIF value > 10) the VIF statistic for each continuous predictor involved in multicollinarity is sent to an output table for further exploration. Model selection success can also be influenced by model specification error when significant higher order model terms (quadratic and cross-product) are omitted from the mixed model. The need for an quadratic term or an interaction between any two predictor variables could be evaluated in the 'quadratic or interaction detection plot respectively. To detect the need for a significant quadratic term, first fit the full model including the quadratic term for the given predictor variable (X i )and examine the Type III P-value for statistical significance and output the predicted values (YHAT full ) for the full model. Then drop both the linear and the quadratic terms for this given predictor from the model and estimate the predicted values for this reduced model (YHAT red ). Then a graphical display between the )yhat (YHAT full -YHAT red ) and X i can reveal the nature and the strength of quadratic effects (Figure 11). Similarly, to detect the need for a significant interaction term between two predictors, first fit the full model including the cross-product term for the two predictor variable (X 1 and X 2 ) and examine the Type III P-value for statistical significance of the interaction term and output the predicted values (YHAT full ) for the full model. Then drop the cross product from the model and estimate the predicted values for this reduced model (YHAT red ). Then a 3-D graphical display between the )yhat (YHAT full -YHAT red ) and X 1 and X 2 can reveal the nature and the strength of interaction effects (Figure 12). Refer to the ALLMIXED macro help file available from the authors website for more information regarding inputting appropriate parameters in the macrocall window. Step5: Final covariance type selection.(optional step- for repeated measures model) After several runs of all possible model selection steps, many data exploration steps, and multicollinarity checks, we can select the final fixed effect model. But, before finalizing the final mixed effect model it is important to verify whether the covariance type used in the model selection step is still the best type for the selected model. Again user-specified covariance types, can be compared and the final covariance type selection can be made based on )AICC j (AICC j - AICC min) using the REML method of PROC MIXED. Refer to the ALLMIXED macro help file available from the authors website for more information regarding inputting appropriate parameters in the macro-call window.
8 Conference On Applied Statistics In Agriculture 153 Step5: Complete mixed model analysis After selecting the final repeated measures mixed model dimensions, complete mixed model analysis can be performed including the data exploration step, mixed model analysis, Lsmean comparisons, model predictions, checking for normality of studentized conditional residuals (Figure 13), and performing influential diagnostics (Figure 14) in one step. Refer to the ALLMIXED macro help file available from the authors website for more information regarding inputting appropriate parameters in the macro-call window. 5. AVAILABILITY OF THE ALLMIXED MACRO: Users can download the ALLMIXED.SAS macro-call file from the authors website at and by clicking the Running puppy dog clip art. Save the ALLMIXED.SAS macro-call file in your PC first and open it in SAS display manager and submit to view the blue macro-call window (You need to have access to INTERNET while running this SAS macro in your system). Input all the required macro input parameters and submit the macro to perform the all possible mixed model selection. Please refer to the required SAS modules listed below for running this macro successfully. Required Sas Modules for Running the All mixed Sas Macro in Version 9.13:! SAS /STAT : PROCS MIXED, CORR, REG and GLMSELECT! SAS/GRAPH: PROC GCHART, PROC GPLOT, PROC G3D! SAS/BASE SAS ODS (RTF, HTML, PDF)! SAS/ACCESS: PC FILES PROC IMPORT and EXPORT 6. SUMMARY The main features of the user-friendly SAS macro application, ALLMIXED are summarized below: The users can input, temporary and permanent SAS data files, Microsoft Excel and Access and comma and TAB delimited text files as input data set. Users can optionally pre-screen the fixed effects and drop obvious non-significant fixed effects if the number of fixed effects exceed 15 using the SAS 9.1 experimental GLMSELECT procedure implemented within the macro. Two new model selection methods, LASSO and LAR are used in this macro to pre-screen the many fixed effects. In case of repeated measures mixed model analysis, the best covariance structure selection from the user specified covariance structures are implemented by comparing the AICC value estimated in the Proc Mixed using REML method and then best covariance structures is graphically identified by searching for the covariance structure with the smallest AICC value. Options for performing all possible fixed effect model selection with and without repeated and random effects and selecting the best candidate models using AICC and MDL estimates using PROC MIXED method ML. In this step, users can differentiate the
9 154 must- keep effects and selectable effects. The all possible model selection process will be performed using the fixed effects identified in the Selectable list of terms. Options are also available for graphical exploration and statistical significance of user specified linear, quadratic, interaction terms for fixed effects. Also, to diagnose multicollinearity (when VIF value > 10) the VIF statistic for each continuous predictors involved in each model selection step are sent to an output table. Also, a boxplot display of VIF estimates by all the continuous fixed effects are generated for the overall assessment of multicollinarity in the model selection process. Options are also available for performing complete mixed model analysis of final model including data exploration, influential diagnostics, and checking for model violations using the experimental ODS GRAPHICS option avilable in Version 9.1. Users can save all SAS output and graphics in Word, HTML, or PDF formats. In addition, full details of all model selection diagnostic statistics are automatically sent to MS excel data tables. SAS log messages are automatically saved to external text log files and only the ERROR and WARNING messages are extracted and displayed as HTML output for easy error checks. Download instructions are given above to download this macro-call file and to perform all possible model selection. 7. REFERENCES Burnham, K. P, and Anderson D. R. (2002) Model selection and inference: a practical information theoretic approach. (Second edition) Springer-Verlag, New York, New York, USA. Cohen R. A (2006) Introducing the GLMSELECT PROCEDURE for Model Selection SUGI31 proceedings Efron, B., Hastie, T., Johnstone, I., and Tibshirani, R. (2004), Least Angle Regression (with discussion),annals of Statistics, 32, Hoeting, J.A, Davis R.A Merton A.A and Thompson S. E (2006) Model selection for Geostatistical Models Ecological Applications, 16(1), pp Keselman, H. J., Algina, J., Kowalchuk, R. K., and Wolfinger, R. D. (1998) A comparison of two approaches for selecting covariance structures in the analysis of repeated measurement, 27, Communications in Statistics, Simulation & Computation Littell, R.C, Milliken, G A., Stroup, WW, and Wolfinger, R D. (2006). SAS System for Mixed Models (second edition), Cary, NC: SAS Institute Inc. Kramer M (2004) Automatic model selection in the mixed model framework KSU applied statistics conference proceedings Ngo, L and Rand, R. (2002). Model Selection in Linear Mixed Effects Models Using SAS Proc Mixed. SUGI 22 SAS Institute (2006) The GLMSELECT procedure (Experimental) Cary, NC Tibshirani, R. (1996) Regression Shrinkage and Selection via the Lasso Journal of the Royal Statistical Society Series B, 58,
10 Conference On Applied Statistics In Agriculture CONTACT INFORMATION Your comments and questions are valued and encouraged. Contact the author: Name: Dr. George C. Fernandez, Enterprise: University of Nevada - Reno Address: CABNR/204 Reno, NV Work phone: (775) gcjf@unr.edu Web: Figure 1 Recommended all possible model selection steps and their sequence
11 156 Figure 2 Selected fixed effects including the continuous and categorical fixed effects based on LASSO selection method in SAS GLMSELECT Figure 3 Selected fixed effects including the continuous and categorical fixed effects based on LAR selection method in SAS GLMSELECT
12 Conference On Applied Statistics In Agriculture 157 Figure 4 Repeated Measure analysis covariance type selection based on smallest AICC. Figure 5 Scatter plot matrix showing rank correlation among 4 AIC and 2 BC based Information criteria statistics in all possible mixed model selection.
13 158 Figure 6 Comparison of penalty factor among AICC, AICC reml and MDL versus number of fixed effect terms. Figure 7 Comparison of the association between AICC and MDL components and the number of fixed effect terms
14 Conference On Applied Statistics In Agriculture 159 Figure 8 Graphical display of best model within each subset and the list of best candidate models identified by the AICC. Figure 9 Graphical display of best model within each subset and the list of best candidate models identified by the MDL
15 160 Figure 10 Box-plot display of VIF statistic of all continuous predictor variables included in the possible model selection. Figure 11 Graphical exploration and the statistical significance of the user-specified quadratic effect.
16 Conference On Applied Statistics In Agriculture 161 Figure 12 Graphical exploration and the statistical significance of the user-specified cross product Figure 13 Mixed model violation detection using studentized conditional residuals
17 162 Figure 14 Repeated measurers mixed model influential diagnostics - at the subject level
All Possible Mixed Model Selection - A User-friendly SAS Macro Application
All Possible Mixed Model Selection - A User-friendly SAS Macro Application George C J Fernandez, University of Nevada - Reno, Reno NV 89557 ABSTRACT A user-friendly SAS macro application to perform all
More informationReal-Time Predictive Modeling of Key Quality Characteristics Using Regularized Regression: SAS Procedures GLMSELECT and LASSO
Paper 8240-2016 Real-Time Predictive Modeling of Key Quality Characteristics Using Regularized Regression: SAS Procedures GLMSELECT and LASSO Jon M. Lindenauer, Weyerhaeuser Company ABSTRACT This paper
More informationAsk the Expert Model Selection Techniques in SAS Enterprise Guide and SAS Enterprise Miner
Ask the Expert Model Selection Techniques in SAS Enterprise Guide and SAS Enterprise Miner SAS Ask the Expert Model Selection Techniques in SAS Enterprise Guide and SAS Enterprise Miner Melodie Rush Principal
More informationPenalizing Your Models: An Overview of the Generalized Regression Platform Michael Crotty, SAS Institute; Clay Barker, SAS Institute
Paper RIV-151 Penalizing Your Models: An Overview of the Generalized Regression Platform Michael Crotty, SAS Institute; Clay Barker, SAS Institute ABSTRACT We provide an overview of the Generalized Regression
More informationPharmaceutical Application for Stability (PASS) Analysis
Pharmaceutical Application for Stability (PASS) Analysis Dr. Vesna Luzar-Stiffler 1,2 and Dr. Charles Stiffler 1 1 CAIR Research Center and 2 University Computing Centre, Zagreb, Croatia Abstract: PASS
More informationEXPERIMENTAL ERROR IN AGRONOMIC FIELD TRIALS
Libraries Conference on Applied Statistics in Agriculture 1994-6th Annual Conference Proceedings EXPERIMENTAL ERROR IN AGRONOMIC FIELD TRIALS Thomas M. Loughin D. F. Cox Paul N. Hinz William T. Schapaugh
More informationRegression diagnostics
Regression diagnostics Biometry 755 Spring 2009 Regression diagnostics p. 1/48 Introduction Every statistical method is developed based on assumptions. The validity of results derived from a given method
More informationGETTING STARTED WITH PROC LOGISTIC
PAPER 255-25 GETTING STARTED WITH PROC LOGISTIC Andrew H. Karp Sierra Information Services, Inc. USA Introduction Logistic Regression is an increasingly popular analytic tool. Used to predict the probability
More informationGasoline Consumption Analysis
Gasoline Consumption Analysis One of the most basic topics in economics is the supply/demand curve. Simply put, the supply offered for sale of a commodity is directly related to its price, while the demand
More informationUse Multi-Stage Model to Target the Most Valuable Customers
ABSTRACT MWSUG 2016 - Paper AA21 Use Multi-Stage Model to Target the Most Valuable Customers Chao Xu, Alliance Data Systems, Columbus, OH Jing Ren, Alliance Data Systems, Columbus, OH Hongying Yang, Alliance
More informationExamination of Cross Validation techniques and the biases they reduce.
Examination of Cross Validation techniques and the biases they reduce. Dr. Jon Starkweather, Research and Statistical Support consultant. The current article continues from last month s brief examples
More informationWeek 11: Collinearity
Week 11: Collinearity Marcelo Coca Perraillon University of Colorado Anschutz Medical Campus Health Services Research Methods I HSMP 7607 2017 c 2017 PERRAILLON ARR 1 Outline Regression and holding other
More informationCONFIDENCE INTERVALS FOR THE COEFFICIENT OF VARIATION
Libraries Annual Conference on Applied Statistics in Agriculture 1996-8th Annual Conference Proceedings CONFIDENCE INTERVALS FOR THE COEFFICIENT OF VARIATION Mark E. Payton Follow this and additional works
More informationGetting Started With PROC LOGISTIC
Getting Started With PROC LOGISTIC Andrew H. Karp Sierra Information Services, Inc. 19229 Sonoma Hwy. PMB 264 Sonoma, California 95476 707 996 7380 SierraInfo@aol.com www.sierrainformation.com Getting
More informationBuilding Better Models with. Pro. Business Analytics Using SAS Enterprise Guide and. SAS Enterprise Miner. Jim Grayson Sam Gardner Mia L.
Building Better Models with SAS Enterprise Miner A JMP Beginner s Pro Guide Business Analytics Using SAS Enterprise Guide and Olivia Parr-Rud Jim Grayson Sam Gardner Mia L. Stephens From Building Better
More informationChecklist before you buy software for questionnaire based surveys
Checklist before you buy for questionnaire based surveys Continuity, stability and development...2 Capacity...2 Safety and security...2 Types of surveys...3 Types of questions...3 Interview functions (Also
More informationAdvanced Tutorials. SESUG '95 Proceedings GETTING STARTED WITH PROC LOGISTIC
GETTING STARTED WITH PROC LOGISTIC Andrew H. Karp Sierra Information Services and University of California, Berkeley Extension Division Introduction Logistic Regression is an increasingly popular analytic
More informationAN AUTOMATED SALES FORECASTING SYSTEM
AN AUTOMATED SALES FORECASTING SYSTEM Sales Forecasting Example 80000 60000 40000 20000 0 Jun-91 Jun-92 Jun-93 Jun-94 Jun-95 Jun-96 Date Italian Italian German Italian German From Bavaria Sales Forecasting
More informationBig Data Executive Program
Big Data Executive Program Big Data Executive Program Data science using Big Data (DS) SAS Business Analytics along with visual analytics brings in the capability to support the business and management
More informationModel Building Process Part 2: Factor Assumptions
Model Building Process Part 2: Factor Assumptions Authored by: Sarah Burke, PhD 17 July 2018 Revised 6 September 2018 The goal of the STAT COE is to assist in developing rigorous, defensible test strategies
More informationGETTING STARTED WITH PROC LOGISTIC
GETTING STARTED WITH PROC LOGISTIC Andrew H. Karp Sierra Information Services and University of California, Berkeley Extension Division Introduction Logistic Regression is an increasingly popular analytic
More informationNew Procedure to Improve the Order Selection of Autoregressive Time Series Model
Journal of Mathematics and Statistics 7 (4): 270-274, 2011 ISSN 1549-3644 2011 Science Publications New Procedure to Improve the Order Selection of Autoregressive Time Series Model Ali Hussein Al-Marshadi
More informationQuantification of Harm -advanced techniques- Mihail Busu, PhD Romanian Competition Council
Quantification of Harm -advanced techniques- Mihail Busu, PhD Romanian Competition Council mihail.busu@competition.ro Summary: I. Comparison Methods 1. Interpolation Method 2. Seasonal Interpolation Method
More informationCorrelation between Carbon Steel Corrosion and Atmospheric Factors in Taiwan
CORROSION SCIENCE AND TECHNOLOGY, Vol.17, No.2(2018), pp.37~44 pissn: 1598-6462 / eissn: 2288-6524 [Research Paper] DOI: https://doi.org/10.14773/cst.2018.17.2.37 Correlation between Carbon Steel Corrosion
More informationAnalysis of Factors Affecting Resignations of University Employees
Analysis of Factors Affecting Resignations of University Employees An exploratory study was conducted to identify factors influencing voluntary resignations at a large research university over the past
More informationRequesting completion of the Work Health Assessment (WHA) Questionnaire. Monitoring completion of the Work Health Assessment Questionnaire
Hiring Offer Phase Contents Page 3 Page 4 Page 5 Page 6-9 Page 10 Page 11-13 Page 14-15 Page 16-17 Page 18-19 Page 20 The Hiring Process Candidate Selection Page - Overview Display Candidates Page - Overview
More informationExploring Treatment By Trial Interaction Using Treatment Stability/Trial Dendrogram Plots
Exploring Treatment By Trial Interaction Using Treatment Stability/Trial Dendrogram Plots Peter Claussen, Steve Gylling Gylling Data Management, Inc. Booth #1020 November 2015 1 Software Overview ARM Single
More informationRegression Analysis I & II
Data for this session is available in Data Regression I & II Regression Analysis I & II Quantitative Methods for Business Skander Esseghaier 1 In this session, you will learn: How to read and interpret
More informationRisk Management User Guide
Risk Management User Guide Version 17 December 2017 Contents About This Guide... 5 Risk Overview... 5 Creating Projects for Risk Management... 5 Project Templates Overview... 5 Add a Project Template...
More informationGenomic Selection with Linear Models and Rank Aggregation
Genomic Selection with Linear Models and Rank Aggregation m.scutari@ucl.ac.uk Genetics Institute March 5th, 2012 Genomic Selection Genomic Selection Genomic Selection: an Overview Genomic selection (GS)
More informationPredictive Modeling Using SAS Visual Statistics: Beyond the Prediction
Paper SAS1774-2015 Predictive Modeling Using SAS Visual Statistics: Beyond the Prediction ABSTRACT Xiangxiang Meng, Wayne Thompson, and Jennifer Ames, SAS Institute Inc. Predictions, including regressions
More informationTutorial #7: LC Segmentation with Ratings-based Conjoint Data
Tutorial #7: LC Segmentation with Ratings-based Conjoint Data This tutorial shows how to use the Latent GOLD Choice program when the scale type of the dependent variable corresponds to a Rating as opposed
More informationLab 1: A review of linear models
Lab 1: A review of linear models The purpose of this lab is to help you review basic statistical methods in linear models and understanding the implementation of these methods in R. In general, we need
More informationAlternative Seasonality Detectors Using SAS /ETS Procedures Joseph Earley, Loyola Marymount University, Los Angeles
Alternative Seasonality Detectors Using SAS /ETS Procedures Joseph Earley, Loyola Marymount University, Los Angeles ABSTRACT Estimating seasonal indices is an important aspect of time series analysis.
More informationComputer Applications for Small Area Estimation, Part 1. Outline
Computer Applications for Small Area Estimation, Part 1 Jerzy Wieczorek Small Area Estimation Research Group, CSRM 1/17/2013 Outline Basic approach for continuous area-level data The estimates we need
More informationPaper Principal Component Regression as a Countermeasure against Collinearity Chong Ho Yu, Ph.D.
Paper 73333 Principal Component Regression as a Countermeasure against Collinearity Chong Ho Yu, Ph.D. ABSTRACT There are different approaches to counteract the threat of multicollinearity in regression
More informationSTP359: Supply Network Inventory in SNC
SAP Training Source To Pay STP359: Supply Network Inventory in SNC External User Training Version: 4.0 Last Updated: 03-Apr-2017 3M Business Transformation & Information Technology Progress set in motion
More informationPractical Aspects of Modelling Techp.iques in Logistic Regression Procedures of the SAS System
r""'=~~"''''''''''''''''''''''''''''\;'=="'~''''o''''"'"''~ ~c_,,..! Practical Aspects of Modelling Techp.iques in Logistic Regression Procedures of the SAS System Rainer Muche 1, Josef HogeP and Olaf
More informationTutorial Segmentation and Classification
MARKETING ENGINEERING FOR EXCEL TUTORIAL VERSION v171025 Tutorial Segmentation and Classification Marketing Engineering for Excel is a Microsoft Excel add-in. The software runs from within Microsoft Excel
More informationSelect2Perform InterView User s Guide
Select2Perform InterView User s Guide Table of Contents INTRODUCTION 1 Licensing 1 Competencies and Custom Questions 1 Interview Guides 1 MANAGING COMPETENCIES 2 Adding a Competency 2 Viewing a Competency
More informationForecasting Construction Cost Index using Energy Price as an Explanatory Variable
Forecasting Construction Cost Index using Energy Price as an Explanatory Variable Variations of ENR (Engineering News Record) Construction Cost Index (CCI) are problematic for cost estimation and bid preparation.
More informationImage is Almost Everything: Displaying Statistics Via ODS R. Scott Leslie, MedImpact Healthcare Systems, Inc., San Diego, CA
Image is Almost Everything: Displaying Statistics Via ODS R. Scott Leslie, MedImpact Healthcare Systems, Inc., San Diego, CA ABSTRACT After conducting statistical analyses you must find a way to report
More informationGetting Started with HLM 5. For Windows
For Windows Updated: August 2012 Table of Contents Section 1: Overview... 3 1.1 About this Document... 3 1.2 Introduction to HLM... 3 1.3 Accessing HLM... 3 1.4 Getting Help with HLM... 3 Section 2: Accessing
More informationEconometric Modeling and Forecasting of Food Exports in Albania
Econometric Modeling and Forecasting of Food Exports in Albania Prof. Assoc. Dr. Alma Braimllari (Spaho) Oltiana Toshkollari University of Tirana, Albania, alma. spaho@unitir. edu. al, spahoa@yahoo. com
More informationFast Track SA 2.0 Drive Thru Timing System. Document: USR-FTSA, Rev. A Effective Date: September 6, 2017
ENGAGE INFLUENCE OPTIMIZE User Guide Fast Track SA 2.0 Drive Thru Timing System Effective Date: September 6, 2017 Fast Track SA 2.0 User Guide Page 2 of 33 Table of Contents 1 INTRODUCTION... 4 2 FAST
More informationNew generation of Operational Lease Management
LeaseProXL New generation of Operational Lease Management Gain a major competitive advantage If you want to drive and inspire your organization, then create a proactive atmosphere by dealing with challenges
More informationI am an experienced SAS programmer but I have not used many SAS/STAT procedures
Which Proc Should I Learn First? A STAT Instructor s Top 5 Modeling Procedures Catherine Truxillo, Ph.D. Manager, Analytical Education SAS Copyright 2010, SAS Institute Inc. All rights reserved. The Target
More informationAssessing the effects of recent events on Chipotle sales revenue
Assessing the effects of recent events on Chipotle sales revenue 1Dr. Simon Sheather SAS Day 2016 2 In February 2005, I moved from Head of the Department of Statistics: March 1, 2005 until February 28,
More informationSAS Enterprise Guide: Point, Click and Run is all that takes
ABSTRACT SAS Enterprise Guide: Point, Click and Run is all that takes Aruna Buddana, TiVo Inc, Alviso, CA The Audience Research and Measurement team at TiVo constantly collects and processes anonymous
More informationDepartment of Economics, University of Michigan, Ann Arbor, MI
Comment Lutz Kilian Department of Economics, University of Michigan, Ann Arbor, MI 489-22 Frank Diebold s personal reflections about the history of the DM test remind us that this test was originally designed
More informationCADE Finance and HR Reports. Administrative Staff Leadership Conference Presenter: Mary Jo Kuffner, Assistant Director Administration
CADE Finance and HR Reports Administrative Staff Leadership Conference Presenter: Mary Jo Kuffner, Assistant Director Administration 1 What Schools and Colleges are Doing to Develop Effective Management
More informationUPIC: Perl scripts to determine the number of SSR markers to run
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Publications from USDA-ARS / UNL Faculty U.S. Department of Agriculture: Agricultural Research Service, Lincoln, Nebraska
More informationThe SPSS Sample Problem To demonstrate these concepts, we will work the sample problem for logistic regression in SPSS Professional Statistics 7.5, pa
The SPSS Sample Problem To demonstrate these concepts, we will work the sample problem for logistic regression in SPSS Professional Statistics 7.5, pages 37-64. The description of the problem can be found
More informationBisan Enterprise. Non Governmental Organizations - NGO Lite Edition. A New Dimension in Financial Management Applications
Bisan Enterprise Non Governmental Organizations - NGO Lite Edition www.bisan.com A New Dimension in Financial Management Applications Bisan Enterprise NGO Lite Edition For small and medium size institutions
More informationSTP358: Order Forecast Monitor in SNC
SAP Training Source To Pay STP358: Order Forecast Monitor in SNC External User Training Version: 4.0 Last Updated: 03-Apr-2017 3M Business Transformation & Information Technology Progress set in motion
More informationUse the AccuSEQ Software v2.0 Mycoplasma SEQ module
QUICK REFERENCE Use the AccuSEQ Software v2.0 Mycoplasma SEQ module Publication Part Number 4425586 Revision Date 25 January 2013 (Rev. B) Mycoplasma SEQ Experiments Note: For safety and biohazard guidelines,
More informationPredictive Modeling using SAS. Principles and Best Practices CAROLYN OLSEN & DANIEL FUHRMANN
Predictive Modeling using SAS Enterprise Miner and SAS/STAT : Principles and Best Practices CAROLYN OLSEN & DANIEL FUHRMANN 1 Overview This presentation will: Provide a brief introduction of how to set
More informationApplying Regression Techniques For Predictive Analytics Paviya George Chemparathy
Applying Regression Techniques For Predictive Analytics Paviya George Chemparathy AGENDA 1. Introduction 2. Use Cases 3. Popular Algorithms 4. Typical Approach 5. Case Study 2016 SAPIENT GLOBAL MARKETS
More informationTo Hydrate or Chlorinate: A Regression Analysis of the Levels od Chlorine in the Public Water Supply
A Regression Analysis of the Levels od Chlorine in the Public Water Supply SAS and all other SAS Institute Inc. product or service names are registered trademarks or trademarks of SAS Institute Inc. in
More informationSAS/STAT 13.1 User s Guide. Introduction to Multivariate Procedures
SAS/STAT 13.1 User s Guide Introduction to Multivariate Procedures This document is an individual chapter from SAS/STAT 13.1 User s Guide. The correct bibliographic citation for the complete manual is
More informationIntroduction to Multivariate Procedures (Book Excerpt)
SAS/STAT 9.22 User s Guide Introduction to Multivariate Procedures (Book Excerpt) SAS Documentation This document is an individual chapter from SAS/STAT 9.22 User s Guide. The correct bibliographic citation
More informationPrimerArray Analysis Tool for Embryonic Stem Cells
For Research Use Manual Table of Contents I. Calculating and exporting Ct values... 3 II. Relative quantification... 4 III. Troubleshooting... 8 2 URL:http://www.takara-bio.com The PrimerArray Analysis
More informationEconometric Analysis Dr. Sobel
Econometric Analysis Dr. Sobel Econometrics Session 1: 1. Building a data set Which software - usually best to use Microsoft Excel (XLS format) but CSV is also okay Variable names (first row only, 15 character
More informationChapter 3. Database and Research Methodology
Chapter 3 Database and Research Methodology In research, the research plan needs to be cautiously designed to yield results that are as objective as realistic. It is the main part of a grant application
More informationVariable Selection Methods
Variable Selection Methods PROBLEM: Find a set of predictor variables which gives a good fit, predicts the dependent value well and is as small as possible. So far have used F and t tests to compare 2
More informationSAS. Data Analyst Training Program. [Since 2009] 21,347+ Participants 10,000+ Brands Trainings 45+ Countries. In exclusive association with
SAS Data Analyst Training Program In exclusive association with 21,347+ Participants 10,000+ Brands 1200+ Trainings 45+ Countries [Since 2009] Training partner for Who Is This Course For? Programmers Non-Programmers
More informationPredicting Corporate 8-K Content Using Machine Learning Techniques
Predicting Corporate 8-K Content Using Machine Learning Techniques Min Ji Lee Graduate School of Business Stanford University Stanford, California 94305 E-mail: minjilee@stanford.edu Hyungjun Lee Department
More informationPrimerArray Analysis Tool Ver. 2.2
For Research Use PrimerArray Analysis Tool Ver. 2.2 Manual Table of Contents I. Calculating and exporting Ct values... 3 II. Relative quantification... 4 III. Troubleshooting...10 2 URL:http://www.takara-bio.com
More informationPrediction and Interpretation for Machine Learning Regression Methods
Paper 1967-2018 Prediction and Interpretation for Machine Learning Regression Methods D. Richard Cutler, Utah State University ABSTRACT The last 30 years has seen extraordinary development of new tools
More informationHierarchical Linear Modeling: A Primer 1 (Measures Within People) R. C. Gardner Department of Psychology
Hierarchical Linear Modeling: A Primer 1 (Measures Within People) R. C. Gardner Department of Psychology As noted previously, Hierarchical Linear Modeling (HLM) can be considered a particular instance
More informationScheduling features for Simulation. Simio User Group /5/2018 Copyright 2017 Simio LLC 1
Scheduling features for Simulation Simio User Group 2018 6/5/2018 Copyright 2017 Simio LLC 1 Simio Product Family Simio Design Edition Simio Team Edition Simio Enterprise Edition Simio Portal Simio Personal
More informationPaper MPSF-073. Model Validity Checks In Data Mining: A Luxury or A Necessity?
Paper MPSF-073 Model Validity Checks In Data Mining: A Luxury or A Necessity? Mike Speed & Simon Sheather Department of Statistics Texas A&M University SAS and all other SAS Institute Inc. product or service
More informationReal-time RT PCR. for identification of differentially expressed genes. (with Schizophrenia application) Rolf Sundberg, Stockholm Univ.
Real-time RT PCR for identification of differentially expressed genes. (with Schizophrenia application) Rolf Sundberg, Stockholm Univ. Göteborg, May 2006 Real-time PCR for mrna quantitation Review paper
More informationBIO 226: Applied Longitudinal Analysis. Homework 2 Solutions Due Thursday, February 21, 2013 [100 points]
Prof. Brent Coull TA Shira Mitchell BIO 226: Applied Longitudinal Analysis Homework 2 Solutions Due Thursday, February 21, 2013 [100 points] Purpose: To provide an introduction to the use of PROC MIXED
More informationPOST GRADUATE PROGRAM IN DATA SCIENCE & MACHINE LEARNING (PGPDM)
OUTLINE FOR THE POST GRADUATE PROGRAM IN DATA SCIENCE & MACHINE LEARNING (PGPDM) Module Subject Topics Learning outcomes Delivered by Exploratory & Visualization Framework Exploratory Data Collection and
More informationFor Demonstrations: Automatic Data Collection So Simple The Graphs Just Appear
For Demonstrations: 800.875.4243 S P C S O F T W A R E S I N C E 1 9 8 3 Automatic Data Collection So Simple The Graphs Just Appear D A T A C O L L E C T I O N / A N A L Y S I S S O F T W A R E QC-CALC
More informationHURST PHENOMENON AND FRACTAL DIMENSIONS IN LONG-TERM YIELD DATA
Libraries Conference on Applied Statistics in Agriculture 1998-10th Annual Conference Proceedings HURST PHENOMENON AND FRACTAL DIMENSIONS IN LONG-TERM YIELD DATA Susanne Aref Follow this and additional
More informationUnit Activity Answer Sheet
Algebra 1 Unit Activity Answer Sheet Unit: Descriptive Statistics The Unit Activity will help you meet these educational goals: Mathematical Practices You will make sense of problems and solve them, reason
More informationLittlefield Labs: Overview
Stanford University Graduate School of Business May 2007 Littlefield Labs: Overview Introduction Littlefield Laboratories is a state-of-the-art highly automated blood samples testing lab. Samples are tested
More informationChapter 3. Basic Statistical Concepts: II. Data Preparation and Screening. Overview. Data preparation. Data screening. Score reliability and validity
Chapter 3 Basic Statistical Concepts: II. Data Preparation and Screening To repeat what others have said, requires education; to challenge it, requires brains. Overview Mary Pettibone Poole Data preparation
More informationStatistical Issues in Meta-Analysis
Statistical Issues in Meta-Analysis In Agricultural Research Fernando E. Miguez Iowa State University Nov 1, 2010 Reviewing Research on Cover Crops Effect of cover crop on yields Issues with individual
More informationMANUAL MY.DHLPARCEL.NL
DHL PARCEL MANUAL MY.DHLPARCEL.NL Log in As soon as we have registered you, you will receive your activation link via e-mail. Log in with your email address, set your own password and start preparing your
More informationNear-Balanced Incomplete Block Designs with An Application to Poster Competitions
Near-Balanced Incomplete Block Designs with An Application to Poster Competitions arxiv:1806.00034v1 [stat.ap] 31 May 2018 Xiaoyue Niu and James L. Rosenberger Department of Statistics, The Pennsylvania
More informationSAS Education Analytic Suite Product Descriptions
SAS Education Analytic Suite Product Descriptions Base SAS Software Base SAS 9.1 is the underlying, fourth generation programming language on which all new SAS applications are written. It is specifically
More informationIBM SPSS Statistics: What s New
: What s New New and enhanced features to accelerate, optimize and simplify data analysis Highlights Extend analytics capabilities to a broader set of users with a cost-effective, pay-as-you-go software
More informationFFH (FINAL FINISH HOST)
FFH by CTI Page 1 05 20 FFH (Final Finish Host) Need a standalone or network solution to meet your QA reporting goals? CTI s Final Finish Host is a software suite developed specifically to manage data
More informationHandy Procedures to Expand Your Analytics Skill Set Mary MacDougall, National City, now part of PNC, Cleveland, OH
Paper B03-2009 Handy Procedures to Expand Your Analytics Skill Set Mary MacDougall, National City, now part of PNC, Cleveland, OH ABSTRACT A good handyman can accomplish a lot with a general-purpose tool
More informationInvoice Printing Users Guide AQUA
Invoice Printing Users Guide AQUA Page 1 of 23 Invoice Printing General Advantage Invoice Printing is designed to provide a centralized area to manage all client invoice printing related activities. Create
More informationEFFICACY OF ROBUST REGRESSION APPLIED TO FRACTIONAL FACTORIAL TREATMENT STRUCTURES MICHAEL MCCANTS
EFFICACY OF ROBUST REGRESSION APPLIED TO FRACTIONAL FACTORIAL TREATMENT STRUCTURES by MICHAEL MCCANTS B.A., WINONA STATE UNIVERSITY, 2007 B.S., WINONA STATE UNIVERSITY, 2008 A THESIS submitted in partial
More informationBiostatistics 208 Data Exploration
Biostatistics 208 Data Exploration Dave Glidden Professor of Biostatistics Univ. of California, San Francisco January 8, 2008 http://www.biostat.ucsf.edu/biostat208 Organization Office hours by appointment
More informationMERCER WIN RELEASE NOTES. November 23, 2013
MERCER WIN RELEASE NOTES November 23, 2013 MERCER WIN RELEASE SUMMARY Release Date 23 November 2013 This Mercer Workforce Intelligence Network (WIN) release incorporates the Mercer International Position
More informationCreative Commons Attribution-NonCommercial-Share Alike License
Author: Brenda Gunderson, Ph.D., 2015 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution- NonCommercial-Share Alike 3.0 Unported License:
More informationSTP351: Purchase Order Collaboration in SNC
SAP Training Source To Pay STP351: Purchase Order Collaboration in SNC External User Training Version: 4.0 Last Updated: 03-Apr-2017 3M Business Transformation & Information Technology Progress set in
More information14 June Access arrangement information for the period 1 July 2017 to 30 June 2022
Attachment 13.1 Analytics + Data Science Report on Methodology for setting the service standard benchmarks and targets Revised proposed access arrangement information 14 June 2018 Access arrangement information
More informationLinear, Machine Learning and Probabilistic Approaches for Time Series Analysis
The 1 th IEEE International Conference on Data Stream Mining & Processing 23-27 August 2016, Lviv, Ukraine Linear, Machine Learning and Probabilistic Approaches for Time Series Analysis B.M.Pavlyshenko
More informationThe SAS System 1. RM-ANOVA analysis of sheep data assuming circularity 2
The SAS System 1 Obs no2 sheep time y 1 1 1 time1 2.197 2 1 1 time2 2.442 3 1 1 time3 2.542 4 1 1 time4 2.241 5 1 1 time5 1.960 6 1 1 time6 1.988 7 1 2 time1 1.932 8 1 2 time2 2.526 9 1 2 time3 2.526 10
More informationModelling and Forecasting the Balance of Trade in Ethiopia
American Journal of Theoretical and Applied Statistics 2015; 4(1-1): 19-23 Published online March 18, 2015 (http://www.sciencepublishinggroup.com/j/ajtas) doi: 10.11648/j.ajtas.s.2015040101.14 ISSN: 2326-8999
More informationBig Data. Methodological issues in using Big Data for Official Statistics
Giulio Barcaroli Istat (barcarol@istat.it) Big Data Effective Processing and Analysis of Very Large and Unstructured data for Official Statistics. Methodological issues in using Big Data for Official Statistics
More informationSoci Statistics for Sociologists
University of North Carolina Chapel Hill Soci708-001 Statistics for Sociologists Fall 2009 Professor François Nielsen Stata Commands for Module 11 Multiple Regression For further information on any command
More informationS-ID Used Subaru Foresters I
S-ID Used Subaru Foresters I Alignments to Content Standards: S-ID.B.6 Task Jane wants to sell her Subaru Forester, but doesn t know what the listing price should be. She checks on craigslist.com and finds
More information